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Minimal Lagrangian connections and related structures

A projective structure on a smooth manifold consists of an equivalence class $$\mathfrak{p}$$ of torsion-free connections on its tangent bundle, where two such connections are called equivalent if they have the same geodesics up to parametrisation.

The representative connections of a projective structure $$\mathfrak{p}$$ on a smooth manifold $$M$$ are in one-to-one correspondence with the sections of an affine bundle $$A \to M$$, whose total space carries a split-signature metric $$h_{\mathfrak{p}}$$ as well as a symplectic form $$\Omega_{\mathfrak{p}}$$, both of which are defined in a canonical fashion from $$\mathfrak{p}$$. Consequently, all the submanifold notions of (pseudo-)Riemannian geometry and symplectic geometry can be applied to the representative connections of $$\mathfrak{p}$$. This point of view gives rise to the notion of a minimal Lagrangian connection.

The aim of this research project is to investigate various questions related to minimal Lagrangian connections, in particular, to provide a characterisation of projective manifolds that arise from a minimal Lagrangian connection.

## Publications

We establish a one-to-one correspondence between Finsler structures on the $$2$$-sphere with constant curvature $$1$$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler $$2$$-spheres of constant curvature and all geodesics closed.

 Journal Journal of the Institute of Mathematics of Jussieu Publisher Cambridge University Press Volume to appear Link to preprint version Link to published version

Related project(s):
68Minimal Lagrangian connections and related structures

We associate a flow $$\phi$$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $$(M,g)$$ of negative Euler characteristic and investigate its properties. We show that $$\phi$$ always admits a dominated splitting and identify special cases in which $$\phi$$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $$(M,g)$$.

 Journal Ergodic Theory and Dynamical Systems Publisher Cambridge University Press Volume 42 Pages 1781--1806 Link to preprint version Link to published version

Related project(s):
68Minimal Lagrangian connections and related structures

Given a parabolic geometry on a smooth manifold $$M$$, we study a natural affine bundle $$A \to M$$, whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on $$A$$, which induces an almost bi-Lagrangian structure on $$A$$ and a compatible linear connection on $$TA$$. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and $$|1|$$-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in $$A$$. For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

 Journal Communications in Contemporary Mathematics Volume to appear Link to preprint version Link to published version

Related project(s):
68Minimal Lagrangian connections and related structures

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